It's a kind of "middle tier" word that is not in frequent use in daily conversation, but is not noticeably unusual either. The use of "despondency" itself is correct in (1) and (2). "felt a feeling" seems faintly repetitious to me (I wonder why not juts "felt despondent"), and I wonder whether the tenses in (1) are exactly in harmony.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.